Continuity and Differentiability | connecting-the-dots

Aspect	Continuity	Differentiability
Definition	No breaks/jumps in the graph	Smooth curve with a defined tangent
Condition	lim⁡ $_{x→a}$ f(x)=f(a)	Derivative exists at x=a
Relationship	Differentiability ⇒ Continuity	Continuity does not ⇒ Differentiability
Example	$(f(x) = x)$ is continuous at 0	$$f(x) =x$$ is continuous at 0

1. Continuity

A function $f$ is said to be continuous at a point $c$ in its domain if the limit of the function at that point equals the value of the function at that point.

Mathematically, $f$ is continuous at $x = c$ if:

$\lim_{x \to c} f(x) = f(c)$

This implies that the Left-Hand Limit (LHL), Right-Hand Limit (RHL), and the function value must all be equal:

$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$

Algebra of Continuous Functions: If $f$ and $g$ are continuous functions at $x = c$ , then:
- $f \pm g$ is continuous at $x = c$ .
- $f \cdot g$ is continuous at $x = c$ .
- $\frac{f}{g}$ is continuous at $x = c$ (provided $g(c) \neq 0$ ).
Composite Functions: If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$ , then the composite function $(f \circ g)$ is continuous at $c$ .

Constant function graph

5.1

Question / Concept	Explanation / Answer	Continuity Status	Differentiability Status
1. Constant Function f(x)=k	Output is always k; graph is a horizontal line.	Continuous for all x.	Differentiable for all x; derivative f′(x)=0.
2. Identity Function f(x)=x	Straight line through origin; slope = 1.	Continuous everywhere.	Differentiable everywhere; f′(x)=1.
3. Polynomial Function f(x)= $x^n$	Smooth curve; no breaks.	Continuous for all real x.	Differentiable for all real x; f′(x)=n  $x^{n−1}$ .
4. Rational Function f(x)=p(x)/q(x)	Continuous except where denominator q(x)=0.	Continuous on domain (where q(x)≠0).	Differentiable on domain; undefined at poles.
5. Root Function f(x)= $x$	Defined only for x≥0.	Continuous for x≥0.	Differentiable for x>0; not differentiable at x=0.
6. Modulus Function (f(x)= \|x\|	V‑shaped graph; sharp corner at 0.	Continuous everywhere	Not differentiable at x=0.
7. Exponential Function f(x)= $e^x$	Smooth increasing curve.	Continuous for all x.	Differentiable for all x; f′(x)= $e^x$
8. Logarithmic Function f(x)=log⁡x	Defined for x>0.	Continuous for x>0.	Differentiable for x>0; f′(x)=1/x.
9. Trigonometric Functions f(x)=sin⁡x,cos⁡x	Periodic and smooth.	Continuous for all x.	Differentiable for all x; f′(x)=cos⁡x,−sin⁡x.
10. Piecewise Functions	Defined by different rules on intervals.	Continuous only if left‑ and right‑limits match at boundaries.	Differentiable only if slopes also match at boundaries.

Function Type	Continuous?	Differentiable?	Notes
Constant	✅ Everywhere	✅ Everywhere	f′(x)=0
Identity	✅ Everywhere	✅ Everywhere	f′(x)=1
Polynomial	✅ Everywhere	✅ Everywhere	Smooth curve
Rational	⚠️ Except where denominator = 0	⚠️ Same restriction	Undefined at poles
Root	✅ for x≥0	⚠️ Not at x=0	Domain limited
Modulus	✅ Everywhere	❌ at x=0	Sharp corner
Exponential	✅ Everywhere	✅ Everywhere	f′(x)= $e^x$
Logarithmic	✅ for x>0	✅ for x>0	Domain limited
Trigonometric	✅ Everywhere	✅ Everywhere	Smooth periodic
Piecewise	⚠️ Depends on matching limits	⚠️ Depends on slope continuity	Check boundaries

What Each Subplot Shows

1 Constant horizontal line f(x)=2 showing continuity everywhere.
2 Identity line f(x)=x continuous and differentiable everywhere.
3 Polynomial f(x)= $x^3$ −3x smooth everywhere.
4 Rational f(x)=1x−1 with a vertical asymptote at x=1.
5 Root f(x)=x domain start at x=0.
6 Modulus f(x)=∣x∣ marking non-differentiable corner at x=0.
7 Exponential f(x)= $e^x$ smooth everywhere.
8 Logarithmic f(x)=ln⁡x domain x>0 shown with boundary near 0.
9 Trigonometric f(x)=sin⁡x periodic and smooth.
10 Piecewise f(x)={x+2} x<0; $x^2$ x≥0 marking the boundary at x=0 and indicating continuity/differentiability status.

Theorem of Limits

Theorem	Formula / Statement	When to use
Constant	lim⁡ $_{x→a}$ c=c	trivial cases
Sum/Product/Quotient	$lim⁡(f±g)=lim⁡f±limg$ , $lim⁡(fg)=lim⁡f⋅lim⁡g$ , $lim⁡(f/g)=lim⁡f/lim⁡g$	component limits exist
Power/Polynomial	lim⁡ $x^n$ = $a^n$ ; polynomials continuous	direct substitution
Composition	lim⁡g(f(x))=g(lim⁡f(x)) if g continuous	nested functions
Squeeze	If h≤f≤g and lim⁡h=lim⁡g=L then lim⁡f=L	bounded/oscillatory cases

2. Differentiability

A function $f$ is differentiable at a point $c$ in its domain if the following limit exists:

$f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$

For differentiability, the Left-Hand Derivative (LHD) must equal the Right-Hand Derivative (RHD).

Relation to Continuity: Every differentiable function is continuous, but the converse is not true. For example, $f(x) = |x|$ is continuous at $x = 0$ but not differentiable there (it has a "sharp corner").

3. Rules of Differentiation

Chain Rule: Used for composite functions. If $y = f(u)$ and $u = g(x)$ , then:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
Implicit Differentiation: When $y$ is not explicitly expressed in terms of $x$ (e.g., $x^2 + y^2 = \sin(xy)$ ), we differentiate both sides with respect to $x$ and solve for $\frac{dy}{dx}$ .
Parametric Differentiation: If $x = f(t)$ and $y = g(t)$ , then:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \quad (\text{provided } \frac{dx}{dt} \neq 0)$

4. Standard Derivatives

Function f(x)	Derivative f′(x)
$\sin^{-1} x$	$\frac{1}{\sqrt{1-x^2}}$
$\cos^{-1} x$	$-\frac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$	$\frac{1}{1+x^2}$
$e^x$	$e^x$
$\log x$	$\frac{1}{x}$
$a^x$	$a^x \log a$

5. Logarithmic Differentiation

This technique is powerful for functions of the form $y = [u(x)]^{v(x)}$ . We take the natural logarithm on both sides ( $\log y = v(x) \log u(x)$ ) and then differentiate implicitly.

6. Second Order Derivative

If $y = f(x)$ , the second order derivative is the derivative of $f'(x)$ , denoted as $\frac{d^2y}{dx^2}$ or $y''$ or $f''(x)$ .

7. Mean Value Theorems

Rolle’s Theorem: If $f$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $f(a) = f(b)$ , then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$ .
Lagrange’s Mean Value Theorem (MVT): If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then there exists at least one $c \in (a, b)$ such that:

$f'(c) = \frac{f(b) - f(a)}{b - a}$

Geometrically, this means there is a point where the tangent is parallel to the secant line joining the endpoints.